On the minimum number of inversions to make a digraph k-(arc-)strong
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The inversion of a set X of vertices in a digraph D consists of reversing the direction of all arcs of D⟨ X⟩. We study sinv'_k(D) (resp. sinv_k(D)) which is the minimum number of inversions needed to transform D into a k-arc-strong (resp. k-strong) digraph and sinv'_k(n) = max{sinv'_k(D) | D }. We show : (i): 1/2log (n - k+1) ≤ sinv'_k(n) ≤log n + 4k -3 ; (ii): for any fixed positive integers k and t, deciding whether a given oriented graph G⃗ satisfies sinv'_k(G⃗) ≤ t (resp. sinv_k(G⃗) ≤ t) is NP-complete ; (iii): if T is a tournament of order at least 2k+1, then sinv'_k(T) ≤ sinv_k(T) ≤ 2k, and sinv'_k(T) ≤4/3k+o(k); (iv):1/2log(2k+1) ≤ sinv'_k(T) ≤ sinv_k(T) for some tournament T of order 2k+1; (v): if T is a tournament of order at least 19k-2 (resp. 11k-2), then sinv'_k(T) ≤ sinv_k(T) ≤ 1 (resp. sinv_k(T) ≤ 3); (vi): for every ϵ>0, there exists C such that sinv'_k(T) ≤ sinv_k(T) ≤ C for every tournament T on at least 2k+1 + ϵ k vertices.
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